# E270 hw 6 | Numerical analysis homework help

NAME:                 Email:                 Enter your LETTER answers HERE               ↓                 1     Note:             2     When done, using your last name and firstname, save THIS FILE as,    3     e270Lastname Firstname HW6 (No space between E270 and Last name) 4     and e-mail it to [email protected]       5                   6     Example:             7     e270Smith Adam HW6 YES       8     e270 Smith Adam HW6 NO       9 10                 11                 12                 13                 14       PAY ATTENTION!     15           16                   17                   18                   19                   20                                       1 Which of the following statements about Type I and Type II errors is correct     a Type I:  Reject a true alternative hypothesis.  Type II: Do not reject a false alternative.   b Type I:  Do not reject a false null hypothesis.  Type II: Reject a true null hypothesis.   c Type I:  Reject a false null hypothesis.  Type II: Reject a true null hypothesis.     d Type I:  Reject a true null hypothesis.  Type II: Do not reject a false null hypothesis.                        2 You are reading a report that contains a hypothesis test you are interested in. The writer of the report writes that the p-value for the test you are interested in is 0.061, but does not tell you the value of the test statistic. From this information you can:     a Not reject the hypothesis at a Probability of Type I error = 0.05, and not reject at a Probability of Type I error = 0.10   b Reject the hypothesis at a Probability of Type I error = .05, and reject at a Probability of Type I error = 0.10   c Not reject the hypothesis at a Probability of Type I error = .05, but reject the hypothesis at a Probability of Type I error = 0.10   d Reject the hypothesis at a Probability of Type I error = .05, but not reject at a Probability of Type I error = 0.10                        3 The random sample below is obtained to test the following hypothesis about the population mean.   H₀:  μ  ≤  120                 H₁:  μ  > 120                 152 203 67 177 220 101 23 214 134   49 53 177 23 128 181 10 103 214   198 99 126 183 70 16 148 118 69   182 166 199 59 172 40 177 28 42   126 104 157 123 199 76 106 162 135   174 55 64 126 176 62 13 59 154   14 196 164 186 71 150 186 90 140   177 189 209 50 26 233 16 28 135   169 171 198 116 115 236 176 80 130   59 227 212 167 35 61 136 72 123   220 100 135 171 70 58 92 28 141   52 27 181 138 231 80 115 153 187   235 212 235 167 136 16 73 166 156   209 128 166 66 234 76 207 154 188   210 202 198 14 192 10 11 136 170   214 231 28 94 125 214 31 64 72                       The level of significance of the test is α = 0.05.   Compute the relevant test statistic.     This is a(n) _______ (two-tail, upper-tail, lower-tail) test.  The test statistic is TS = _______.   a Upper tail test.   TS = 1.34             Do not reject H₀:  μ  ≤ 120.  Conclude that the population mean is not greater than 120.   b Upper tail test.   TS = 1.88             Reject H₀:  μ  ≤ 120.  Conclude that the population mean is greater than 120.     c Upper tail test.   TS = 1.88             Reject H₀:  μ  ≤ 120.  Conclude that the population mean is greater than 120.     d Lower tail test. TS = 1.88             Do not reject H₀:  μ  ≤ 120.  Conclude that the population mean is less than 120.                         4 Consider the following hypothesis test.             H₀:  μ  ≥ 15                 H₁:  μ  < 15                 A random sample of n = 15 yielded the following observations         8 7 11 11 8           13 8 12 13 5           21 21 19 15 18           Use α = 0.05                 TS = ______ CV = ______ State the decision rule.         a -1.743 -1.761 Do not reject H₀.  Conclude the mean is not less than 15.     b -1.74 -1.64 Reject H₀.  Conclude the mean is less than 15.     c 1.847 2.145 Do not reject H₀.  Conclude the mean is not less than 15.     d 1.847 1.761 Reject H₀.  Conclude the mean is less than 15.                         5 In a recent study, a major fast food restaurant had a mean service time of 164 seconds.  The company embarks on a quality improvement effort to reduce the service time and has developed improvements to the service process.  The new process will be tested in a sample of stores.  The new process will be adopted in all of its stores, if it resulted in decreased service time.  To perform the hypothesis test in the previous question, the sample of 54 stores yields the following data (seconds).                               157 115 115 115 134 174 128 136 161   127 139 125 145 199 161 182 144 199   156 117 129 193 173 146 128 166 185   147 136 180 184 116 172 116 193 183   184 160 120 161 161 122 191 170 124   130 191 170 190 194 139 114 195 183                       Use α = 0.05                 |TS| = ______ |CV| = ______             a 2.335 1.674 Do not reject H₀.  The mean is not less than 164 seconds.  Do not adopt the new process.       b 2.335 1.674 Reject H₀.  The mean is less than 164 seconds.  Adopt the new process. c 1.674 1.349 Reject H₀.  The mean is less than 164 seconds.  Adopt the new process. d 1.349 1.674 Do not reject H₀.  The mean is not less than 164 seconds.  Do not adopt the new process.                           6 According to Kelley Blue Book, the mean price for one-to three-year-old used cars nationwide is \$23,400.  to compare the average price of similar used cars in central indiana, a random sample of 120 such cars were selected.  The sample mean was \$21,824 with a standard deviation of 7,309.  Does the sample provide significant evidence that the mean price of one-to-three-year old used cars is different from the national mean price?           Use α = 0.05               a p-value = 0.044 Reject H₀.  Conclude that the dealership’s price is different from the national mean price.       b p-value = 0.124 Do not reject H₀.  Conclude that the dealership’s price is not different from the national mean price.       c p-value = 0.0091 Do not reject H₀.  Conclude that the dealership’s price is not different from the national mean price.       d p-value = 0.0182 Reject H₀.  Conclude that the dealership’s price is different from the national mean price.                           7 The 2009 mean annual salary of business degree graduates in accounting was \$47,900.  In a follow-up study in June 2011, a sample of n = 120 graduating accounting majors yielded a sample mean of \$49,500 and standard deviation of \$8,200.  Does the 2011 study provide a significant proof that the mean salary in 2011 is higher than in 2009?  Perform this test of hypothesis at a 5% level of significance.       a p-value = 0.0524 Do not reject H₀.  Conclude that the mean annual salary in 2011 is no greater than in 2009.       b p-value = 0.0524 Reject H₀.  Conclude that the mean annual salary in 2011 is greater than in 2009. c p-value = 0.0162 Reject H₀.  Conclude that the mean annual salary in 2011 is greater than in 2009. d p-value = 0.0162 Do not reject H₀.  Conclude that the mean annual salary in 2011 is no greater than in 2009.                           8 A production line operates with a mean filling weight of 16 ounces per container.  Overfilling or under filling is a serious problem, and the production line should be shut down if either occurs.  A quality control inspector samples 20 items every 2 hours and at that time makes the decision of whether to shut the line down for adjustment.  On sample provides the following data:           15.8 16.1 16.2 16.1 16.1           16.6 16.3 16.3 15.9 16.1           16.2 15.8 16 16.3 15.9           16 15.9 16.1 15.9 16.2         α = 0.05                 Decision Rule:  Reject H₀ if TS > CV             TS = ______ CV = ______               a 2.000 2.093 Do not reject H₀.  Do  not shut the line down for adjustment.   b 2.000 1.96 Reject H₀.  Shut the line down for adjustment.     c 1.786 1.729 Reject H₀.  Shut the line down for adjustment.     d 2.04 1.64 Do not reject H₀.  Shut the line down for adjustment.                         9 The mean cholesterol level in women ages 21-40 in the United States is 190 mg/dl.  A study is conducted to determine the cholesterol levels among recent female Asian immigrants.  The following is the cholesterol level of a random sample of 108 recent female Asian immigrants.       239 105 251 216 220 120 218 195 129   125 196 193 108 178 187 111 176 178   141 190 214 180 172 204 118 108 124   238 248 253 208 135 146 122 209 254   209 232 238 251 110 224 249 219 219   124 226 252 189 212 163 205 202 190   195 116 125 250 244 140 237 192 191   224 105 201 194 136 245 118 150 165   132 171 245 166 218 159 130 255 131   185 210 223 153 167 174 239 200 107   235 123 224 221 106 212 212 130 154   200 140 170 202 247 112 153 150 205   Does the sample provide significant evidence that mean cholesterol level of recent female Asian immigrants is lower than the mean cholesterol level among all females in the United States?  State the null and alternative hypotheses.  Compute the test statistic and the p-value.  State the decision rule.       Round x̅ to two decimal points and the standard error to three decimal points.       p-value = ______               a 0.069 The evidence is significant at α = 0.05, but not significant at α = 0.01.   b 0.069 The evidence is significant at α = 0.10, but not significant at α = 0.05.   c 0.034 The evidence is significant at α = 0.05, but not significant at α = 0.01.   d 0.034 The evidence is significant at α = 0.01, but not significant at α = 0.05.                       10 We want to test the hypothesis that mothers with low socio-economic status (SES) deliver babies whose birth weights are lower than “normal”.  To test this hypothesis, a list is obtained of birth weights from 100 consecutive, full-term, live-born deliveries from the maternity ward of a hospital in a low-SES area.  The mean birth weight is x̅ = 115 oz. with a standard deviation s = 24 oz.  Nationwide, the mean birth weight in the United States is 120 oz.  At α = 0.05, does this sample provide significant evidence that the mean birth weight of babies born to mother with low SES is lower than “normal”?           a p-value = 0.0188 Reject H₀ at the 5 percent level of significance.  Conclude that the mean birth weight of babies born to low-SES mothers is lower than “normal”.       b p-value = 0.0188 Reject H₀ at the 1 percent level of significance.  Conclude that the mean birth weight of babies born to low-SES mothers is lower than “normal”.       c p-value = 0.0785 Do not reject H₀ at the 5 percent level of significance.  Conclude that the mean birth weight of babies born to low-SES mothers is no lower than “normal”.       d p-value = 0.0785 Do not reject H₀ at the 10 percent level of significance.  Conclude that the mean birth weight of babies born to low-SES mothers is no lower than “normal”.                           11 At Western University the historical mean scholarship examination score of entering students has been 900.  Each year a sample of applications is taken to see whether the examination scores are at the same level as in previous years.  The null hypothesis tested is H₀: μ = 900.  A sample of n = 81 students in this year’s class provided a sample mean score of x̅ = 935 and a standard deviation of s = 180.          First build a 95% confidence interval for the population mean score.       a The confidence interval captures µ₀ = 900.  Do not reject H₀.  Conclude that the current mean score is not different from the historical mean score.   b Compared to the MOE, x̅ − µ₀ is within the margin of error.  Conclude that the current mean score is not different from the historical mean score.   c Compared to the MOE, x̅ − µ₀ is outside the margin of error.  Conclude that the current mean score is different from the historical mean score.   d Both a and b are correct.                                 12 Consider the following hypothesis test.               H₀:  π  ≤ 0.5 H₁:  π  > 0.5           A sample of n = 200 provided a sample proportion of p̅ = 0.57.  At α = 0.05, what is your conclusion?   TS = ______ CV = ______ State the decision rule.         a 1.98 1.64 Conclude the population proportion is no greater than 0.50.   b 1.98 1.64 Conclude the population proportion is greater than 0.50.   c 2.98 1.96 Conclude the population proportion is no greater than 0.50.   d 2.98 1.96 Conclude the population proportion is greater than 0.50.                       13 In the previous question, the prob value for the test is:         a 0.0239                 b 0.0427                 c 0.0618                 d 0.0808                                     Next THREE questions are based on the following         Consider the following hypothesis test.                   H₀:  π ≥ 0.75                 H₁:  π < 0.75           Compute the test statistic and the p-value for the following three cases.       14 n = 200 p̅ = 0.70 α = 0.05       a p-value = 0.0258 Conclude that the population proportion is not less than 0.75.   b p-value = 0.0258 Conclude that the population proportion is less than 0.75.   c p-value = 0.0516 Conclude that the population proportion is not less than 0.75.   d p-value = 0.0516 Conclude that the population proportion is less than 0.75.                       15 n = 200 p̅ = 0.70 α = 0.10       a p-value = 0.0516 Conclude that the population proportion is less than 0.75.   b p-value = 0.0516 Conclude that the population proportion is not less than 0.75.   c p-value = 0.0258 Conclude that the population proportion is less than 0.75.   d p-value = 0.0258 Conclude that the population proportion is not less than 0.75.                       16 n = 900 p̅ = 0.72           a p-value = 0.0188 Reject H₀ at α = 0.10, but do not reject at α = 0.05.     b p-value = 0.0188 Reject H₀ at α = 0.05, but do not reject at α = 0.01.     c p-value = 0.0672 Reject H₀ at α = 0.10, but do not reject at α = 0.05.     d p-value = 0.0672 Reject H₀ at α = 0.05, but do not reject at α = 0.10.                         17 The Center for Workforce Development found that 40% of Internet users received more than 15 e-mail messages per day in 2008.  In 2012, a similar study on the use of e-mail was repeated.  The purpose of the study was see whether use of e-mail has increased.  Formulate the null and alternative hypotheses to determine whether an increase has occurred in the proportion of Internet users receiving more than 10 e-mail messages per day.           To test the hypothesis at a 5% level of significance, a sample of 420 Internet users found 189 receiving more than 10 e-mail messages per day.  Compute the test statistic and the p-value.     p-value = ______.               a 0.0183 Reject H₀ at α = 0.10.  Conclude the population proportion is greater than 0.40.   b 0.0183 Reject H₀ at α = 0.05.  Conclude the population proportion is greater than 0.40.   c 0.0544 Do not reject H₀ at α = 0.05.  Conclude the population proportion is not greater than 0.40. d Both a and b are correct.                                 18 We want to test the hypothesis that at least 75% of drivers on a freeway violate the speed limit.  In a random sample of n = 900 vehicles, 657 violated the speed limit.  Compute the sample proportion.     State the null and alternative hypotheses and the decision rule.       a Reject H₀ at α = 10% and conclude less than 75% of drivers violate the speed limit.  But, do not reject H₀ at α = 5% and conclude 75% or more of drivers violate the speed limit.   b Reject H₀ at α = 5%.  Conclude less than 75% of drivers violate the speed limit.     c Reject H₀ at α = 5%.  Conclude more than 75% of drivers violate the speed limit.   d Reject H₀ at α = 1%.  Conclude less than 75% of drivers violate the speed limit.                         19 At least 20% of all workers are believed to be willing to work fewer hours for less pay to obtain more time for personal and leisure activities.  A recent poll consisting of 600 respondents found 17% willing to work fewer hours for less pay to obtain more personal and leisure time.  At 5% level of significance, does the sample result support the claim that at least 20% of all workers are willing to work fewer hours for less pay to obtain more time for personal and leisure activities? Round the proportion to two decimal point.           a 1.64 1.84 Do not reject H₀.  Conclude that no less than 20% are willing to work fewer hours for less pay to obtain more time for personal and leisure activities.       b 1.84 1.96 Do not reject H₀.  Conclude that no less than 20% are willing to work fewer hours for less pay to obtain more time for personal and leisure activities.       c 1.84 1.64 Reject H₀.  Conclude that less than 20% are willing to work fewer hours for less pay to obtain more time for personal and leisure activities.       d 1.56 1.64 Do not reject H₀.  Conclude that no less than 20% are willing to work fewer hours for less pay to obtain more time for personal and leisure activities.                           20 In the previous question, the p-value is _______.         a 0.0594                 b 0.0329                 c 0.0233                 d 0.0158
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